Sometimes R built-in statistical test functions such as cor.test report a p-value of 0, which makes me wonder the smallest p-value that can precisely be returned by these functions. I think this should be smaller that 1e-21, because I've seen something around 3e-21 returned.
This thread might help address your question: https://stackoverflow.com/questions/35349388/precision-of-scientific-notation-in-r
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You must have a very special kind of question to be worrying about numbers like 3e-21.
There are sources of uncertainty in p-values that don't stem from computer floating-point arithmetic. See, for example, this paper.
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